The exhaust gas emitted from an internal combustion engine is a heterogeneous mixture that contains gaseous emissions such as carbon monoxide (“CO”), unburned hydrocarbons (“HC”) and oxides of nitrogen (“NOx”) as well as condensed phase materials (liquids and solids) that constitute particulate matter (“PM”). An exhaust treatment technology in use for high levels of particulate mater reduction may include a particulate filter (“PF”) that traps particulate matter. Regeneration is the process of removing the accumulated particulate matter from the PF. However, uncontrolled regeneration may occur during certain operation conditions. Specifically, if the engine speed drops to idle during regeneration, the exhaust gas flowing through the PF will significantly decrease, while at the same time the concentration of oxygen in the PF will increase. Because the regeneration that is in progress creates an elevated substrate temperature of the PF, the combination of decreased flow rate and increasing oxygen concentration may create an uncontrolled reaction that elevates the PF to a higher temperature. Such high temperature gradients tend to increase the stress of the PF. Repeated thermal shock may create a cumulative effect that may eventually lead to cracking of the substrate of the PF. In some instances, even a single drop-to-idle event may potentially create a temperature gradient that cracks the substrate of the PF.
Two stage or three stage regeneration of a PF takes place when the temperature set point of the exhaust gas entering the PF is raised in increments depending on the temperature and the soot loading of the PF. However, multi-stage regeneration does not take into account that soot loading of the PF will continuously change during regeneration. Thus, because the temperature set point is set based on the soot loading before regeneration, the temperature of the PF during regeneration is typically lower than what is needed for high regeneration efficiency. Moreover, the substrate of the PF has a stratified temperature, where the temperature gradually decreases from the center to the outer surface of the PF. The stratified temperature of the PF results in different soot burning rates throughout the PF. Multi-stage regeneration does not take into account the stratified temperature of the PF substrate. Accordingly, it is desirable to provide an efficient approach to regenerate a PF, while at the same time minimizing the temperature gradient and risk of uncontrolled regeneration in the PF.